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Course: Calculus 2 > Unit 3

Lesson 6: Separation of variables

Separable differential equations

Separation of variables is a common method for solving differential equations. Learn how it's done and why it's called this way.

Separation of variables is a common method for solving differential equations. Let's see how it's done by solving the differential equation

\frac{d y}{d x} = \frac{2 x}{3 y^{2}}

\begin{aligned} (1) & \frac{d y}{d x} & = \frac{2 x}{3 y^{2}} \\ (2) & 3 y^{2} \cdot \frac{d y}{d x} & = 2 x & Multiply by 3 y^{2} \\ (3) & 3 y^{2} d y & = 2 x d x & Multiply by d x \\ (4) & \int 3 y^{2} d y & = \int 2 x d x & Take integral \\ (5) & y^{3} & = x^{2} + C & Integrate \\ (6) & y & = \sqrt[3]{x^{2} + C} & Isolate y \end{aligned}

Let's review this solution.

In rows

(1)

(3)

we manipulated the equation so it was in the form

f (y) d y = g (x) d x

. In other words, we separated

x

and

y

so each variable had its own side, including the

d x

and the

d y

that formed the derivative expression

\frac{d y}{d x}

. This is why the method is called "separation of variables."

In row

(4)

we took the indefinite integral of each side of the equation. The underlying principle, as always with equations, is that if

f (y) d y

is equal to

g (x) d x

, then their indefinite integrals must also be equal.

In rows

(5)

and

(6)

we performed the integration with respect to

y

(on the left-hand side) and with respect to

x

(on the right-hand side) and then isolated

y

We only added a constant

C

on the right-hand side. Adding a constant to both sides would be unnecessary, because we can then move one of the constants to the other side and end up with a single constant.

In conclusion, the general solution of

\frac{d y}{d x} = \frac{2 x}{3 y^{2}}

y = \sqrt[3]{x^{2} + C}

. You can differentiate

y

to verify this solution.

Looking back at the equation's solution, notice how the separation of variables that we performed in rows

(1)

(3)

allowed us to integrate each side and obtain an equation without a derivative.

Problem 1.A

Problem set 1 will walk you through the process of solving this differential equation:

\frac{d y}{d x} = e^{x} \cdot y^{2}

How does the equation look after the separation of variables?

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